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In the 1923 note Waves and Quanta, Louis de Broglie proposed a periodic process associated with rest energy, remaining in phase with a wave along the moving particle's path. He distinguished the internal frequency slowed by time dilation from the wave frequency based on total energy.^[ref] The proposal developed into his 1924 thesis.^[ref] In the 1929 lecture The wave nature of the electron, he reconstructed the argument using a Lorentz transformation.^[ref]

Here is that quantum phase argument in modern notation. Set ${\omega }_0=m{c}^2/\hslash$ in the rest frame. A Lorentz transformation to a frame where the particle has velocity $v$ givesAlong $x=vt$, proper time is $\tau =t/\gamma$, soThis is the harmony: evaluating the spatial wave at the moving particle gives the slower clock phase. At fixed $x$, its angular frequency is instead $E/\hslash =\gamma {\omega }_0$. Since $k=p/\hslash$, the de Broglie wavelength follows as $\lambda =2\pi /k=h/p$. The proposed clock was a theoretical construction, not an observed mechanical vibration.

Louis de Broglie proposed associating a wave with material particles, relating its wavelength to the particle momentum by $\lambda =h/p$. This connects a particle quantity with an experimentally testable wave scale: increasing the momentum decreases the expected diffraction wavelength. His account of matter waves helped motivate a description in which standing-wave conditions select allowed atomic motions, and his Nobel lecture discusses the relation between this proposal and later electron-diffraction experiments.^[ref]

Theoretical physics.^[ref]

Awarded to Louis de Broglie for the wave nature of electrons. The de Broglie wavelength connects momentum to observable diffraction.^[ref]